A Shrinking Core Model for Steam Hydration of CaO-Based Sorbents Cycled for CO2 Capture: Supplementary Information

A Shrinking Core Model for Steam Hydration of CaO-Based Sorbents Cycled for CO2 Capture: Supplementary Information John Blamey 1, Ming Zhao 2, Vasilije Manovic 3, Edward J. Anthony 3, Denis R. Dugwell 1, Paul S. Fennell 1 * 1 Department of Chemical Engineering, Imperial College London, South Kensington, London, SW7 2AZ, UK 2 School of Environment, Tsinghua University, Beijing 100084, China 3 Cranfield University, Cranfield, Bedfordshire MK43 0AL, United Kingdom *Corresponding author. Tel: +44 (0 20 7594 6637. Fax: +44 (0 20 7594 5638. E-mail: p.fennell@imperial.ac.uk Nomenclature C Molar concentration mol/m 3 D e Effective diffusivity within pores m 2 /s D g Gas-phase diffusivity m 2 /s D K Knudsen diffusivity m 2 /s d p Particle diameter m d pore Pore diameter m k A First order rate constant for the reaction of CaO with steam m/s k B Boltzmann constant J/K k g Mass transfer coefficient m/s M Molar mass g/mol

N Number of cycles n Number of moles mol p Pressure Pa R Universal gas constant J/mol/K Re Reynolds number r Radius m S V Specific, BET, surface area, expressed in m 2 /m 3 m 2 /m 3 Sc Sh Schmi number Sherwood number T Temperature K t Time s U Fluid velocity m/s U mf Minimum fluidisation velocity m/s W Molar flux mol/m 2 /s X Mole fraction X Ca(OH2 Conversion to Ca(OH 2 ΔH r θ Enthalpy of reaction under standard conditions kj/mol ε x Porosity of species x λ Mean free path m μ g Fluid viscosity kg/m/s ρ g Gas density kg/m 3 ρ x Density of species x kg/m 3 σ Collision diameter m τ pore Ω Pore tortuosity Collision integral

Subscripts for C, r, X c/c s/s B E At the core of the particle CaO/Ca(OH 2 interface At the surface of particles In the bulk phase (not applicable to r At equilibrium (not applicable to r 1 Bulk Diffusivity and Mass Transfer Coefficients The gas-phase diffusivity (D g was calculated from Chapman-Enskog theory (see Table 1, using Eq. 1, taken from Cussler [1] and adapted for SI units, where T is the temperature of hydration, M x the molar mass of species x and p the bulk pressure. D g 5.96 10 24 T 3/2 (1/M H2 O + 1/M N2 1/2 pσ 2 12 Ω Eq. 1 The collision diameter (σ 12 was calculated as the arithmetic mean of the two species, σ 12 0.5(σ H2 O + σ N2, and the collision integral (Ω was calculated as Ω f(k B T/ε 12 using a correlation provided by Cussler [1], where k B is the Boltzmann constant and the energy of interaction (ε 12 was calculated as the geometric mean of the two species, ε 12 (ε CO2 ε N2 1/2. All collision diameters and energies of interaction were provided by Cussler [1]. The Sherwood Number (Sh for this work has been estimated (see Table 1 using a correlation from Perry and Green [2], Sh 0.91Re 0.49 p Sc 0.33, which is dependent on the Reynolds (Re and Schmi (Sc Numbers, defined by Re p Ud p ρ g /μ g and Sc μ g /ρ g D g respectively. The fluid viscosities (μ g and densities (ρ g were obtained from the NIST Chemistry Webbook [3] and the fluid velocity (U was calculated for the

temperatures of interest (473, 573 and 673 K using a cold (293 K inlet flow-rate of 100 cm 3 /s and an internal diameter of the TGA furnace of 17 mm. An estimate of the mass transfer coefficient (k g was calculated (see Table 1 for the temperatures of interest from the Sherwood Number, gas-phase diffusivity and the particle diameter (d p, as k g D g Sh/d p. Table 1 Calculated values of gas phase diffusivities, Sherwood Numbers and mass transfer coefficients at the temperatures of investigation Temperature of hydration [K] 473 573 673 Gas phase diffusivity [m 2 /s] 5.15x10-5 7.30x10-5 9.73x10-5 Sherwood Number 1.78 1.64 1.53 Mass transfer coefficient [m/s] 0.154 0.201 0.249 The mass transfer coefficient can be used to calculate the flux of H 2O at the surface of the particle (W H2 O,r s, with knowledge of the molar bulk and surface concentrations of H 2O (C B and C S respectively, using Eq. 2. Assuming a perfect gas and knowledge of the overall molar concentration (C, the mole fractions of H 2O can be calculated from C x CX x. W H2 O,r s k g (C B C S k g C(X B X S Eq. 2 2 Intra-Particle Diffusivity The mechanism of diffusion through the particle has to be established, in order to calculate the intra-particle diffusivity. To establish whether Knudsen diffusion contributes, the mean free path of molecules has to be compared to the average pore diameter; Knudsen diffusion occurs when molecules collide with pore walls more frequently than with other molecules. The mean free path (λ was calculated, using λ 1/ 2πσ H2 OC, as 208, 252 and 296 nm for 473, 573 and 673 K respectively. The pore size distribution of the Ca(OH 2 product layer through which the steam diffuses to react with the CaO was not directly measured. Therefore, it was calculated indirectly,

using data obtained for porosity (obtained from mercury porosimetry measurements, < 360 μm and average pore diameter (as obtained from nitrogen adsorption studies for the cycled CaO. First, the porosity of Ca(OH 2 was calculated from a mass balance of CaO and Ca(OH 2, assuming no change in particle/layer size (see Eq. 3. Then, the average pore size was calculated, assuming a constant pore length (see Eq. 4, i.e., constant growth along the length of the pore. Experimental data for CaO and calculated data for Ca(OH 2 are presented in Table 2. ε Ca(OH2 1 ρ CaOM Ca(OH2 (1 ε CaO ρ Ca(OH2 M CaO Eq. 3 d pore,ca(oh2 d pore,caoε Ca(OH2 ε CaO Eq. 4 Table 2 Experimental data for CaO and calculated data for Ca(OH 2 Cycle CaO porosity CaO average pore Ca(OH 2 porosity Ca(OH 2 average number diameter [nm] pore diameter [nm] 0 0.685 25.6 0.372 13.9 2 0.597 34.2 0.196 11.2 6 0.566 36.0 0.134 8.6 13 0.553 34.0 0.109 6.7 The pore diameter is therefore smaller than the mean free path length (7-14 nm vs. 210-300 nm. The Knudsen Numbers (λ/d pore are between 15 and 45 depending on extent of cycling and temperature of hydration, which are within the range in where Knudsen diffusion is expected. The Knudsen diffusivity (D K was calculated according to Eq. 5 (see Table 3, using the molar mass of H 2O [1]. Then, the effective diffusivity (D e through the pores was derived according to Eq. 6 using the conventional expression for resistances in series [4] combined with a term to account for porosity and pore tortuosity [1] (see

Table 3. The pore tortuosity (τ pore, which is a factor to account for the non-linear nature of pores, was taken as 3 [1]. This is a typical value suggested, in absence of experimental data, by Cussler [1]. D K 1 3 d pore ( 2RT 1/2 Eq. 5 πm H2 O D e ε Ca(OH 2 ( 1 + 1 1 Eq. 6 τ pore D g D K Table 3 Calculated Knudsen and effective diffusivity for hydration temperatures and cycling extents investigated Cycle Knudsen diffusivity [m 2 /s] Effective diffusivity [m 2 /s] Number at 473 K at 573 K at 673 K at 473 K at 573 K at 673 K 0 3.45x10-6 3.80x10-6 4.11x10-6 4.00x10-7 4.47x10-7 4.89x10-7 2 2.79x10-6 3.08x10-6 3.33x10-6 1.73x10-7 1.93x10-7 2.11x10-7 6 2.13x10-6 2.34x10-6 2.54x10-6 9.16x10-8 1.02x10-7 1.11x10-7 13 1.66x10-6 1.83x10-6 1.98x10-6 5.84x10-8 6.48x10-8 7.05x10-8 3 Mass Transfer through the Product Layer A mass balance on the flux of H 2O (W H2 O between r and Δr in the product layer, assuming quasisteady state conditions, results in Eq. 7 [5]. d(w H2 O,rr 2 0 dr Eq. 7 The constitutive equation (or Fick s Law for a constant molar concentration is given by Eq. 8 [6], where the first term refers to convective flux, the second to diffusive flux, and species A and B are the diffusing and counter-diffusing gases respectively. For the hydration of lime, species A is H 2O and there is no counter-diffusing gas; therefore Eq. 8 rearranges to become Eq. 9.

W Ar (W Ar + W Br X A CD e dx A dr W H2 O,r CD e dx H2 O,r (1 X H2 O,r dr Eq. 8 Eq. 9 Combining Eq. 7 and Eq. 9 gives Eq. 10. d dr ( CD er 2 dx H2 O,r 0 (1 X H2 O,r dr Eq. 10 To solve for X H2 O,r in this Eq. 10, the boundary conditions must be applied: at the first boundary condition (BC1 r r S and X H2 O,r X S, and at the second (BC2 r r C and X H2 O,r X C. X H2 O 1 (1 X C [ (1 X 1 ( S (1 X C ] r 1 Eq. 11 ( 1 1 r C r S r C X H2 O from Eq. 11 is then differentiated with respect to r to give Eq. 12. dx H2 O dr (1 X C [ (1 X 1 ( S (1 X C ] r 1 r 2 ( 1 1 r S r C r ( 1 C r 1 S r C ln [ (1 X S (1 X C ] Eq. 12 Substituting Eq. 11 and Eq. 12 into Eq. 9 gives Eq. 13, the equation for molar flux through the product layer. W H2 O,r CD e ln [ (1 X S (1 X C ] r 2 ( 1 r S 1 r C Eq. 13

4 Mass Balance on Ca(OH2 Formation during Reaction A balance on the number of moles of CaO at time t (n CaO,t is given by Eq. 14. Noting that (i the rate of production of Ca(OH 2 is the negative rate of production of CaO and (ii the rate of production of Ca(OH 2 is also the negative rate of production of H 2O, the rate of production of Ca(OH 2 is given by Eq. 15. From this, Eq. 16 for the rate of change of the core radius can be derived. n CaO,t 4πr c,t 3 ρ CaO (1 ε CaO Eq. 14 3M CaO dn Ca(OH2 dn CaO dr C M CaOW H2 O,r c ρ CaO (1 ε CaO d ( 4πr c 3 ρ CaO (1 ε CaO 3M CaO dn H 2 O 4πr C 2 W H2 O,r c Eq. 15 Eq. 16 A constant external diameter of particles has been assumed; therefore, r C can be related to the conversion to Ca(OH 2 (X Ca(OH2 within the particles according to Eq. 17. X Ca(OH2 1 ( r 3 c Eq. 17 r s By differentiating X Ca(OH2 in Eq. 17 with respect to r C and multiplying by dr c / in Eq. 16 (i.e., by applying the chain rule, and substituting for r c /r s, Eq. 18 can be derived. This gives a relationship between experimentally derived rates of reaction and the flux of H 2O at the interface of the core and the product layer. dx Ca(OH2 3 r s (1 X Ca(OH2 2/3 M CaO W H2 O,r c ρ CaO (1 ε CaO Eq. 18

5 First Order Kinetics of Reaction First order reaction kinetics is assumed and therefore the rate of formation of Ca(OH 2 is given in Eq. 19, where k A is the first order rate constant per unit surface area, S V is the BET surface area per unit volume (using data presented in Table 1 [see full paper], expressed in m 2 /m 3, calculated from S V S BET ρ CaO (1 ε CaO and X C and X E are the mole fraction of H 2O at the core and the mole fraction associated with the equilibrium partial pressure of H 2O over Ca(OH 2 at the temperature of hydration respectively. (X C X E is chosen as the effective mole fraction of H 2O driving the reaction; note that the reaction will cease if X C X E. d(n Ca(OH2 4πr c 3 S V k 3 A C(X C X E Eq. 19 The molar rate of reaction of Ca(OH 2 is related to the flux of H 2O at the interface of the core and the product layer using Eq. 20; it therefore follows that the flux of H 2O at the interface can be presented by Eq. 21. W H2 O,r c 1 d(n Ca(OH2 4πr2 c W H2 O,r c r cs V k 3 A C(X C X E Eq. 20 Eq. 21 6 Application of the Shrinking Core model The application of the model proceeded as follows: 6.1 Fundamental Equations To recap, the fundamental equations are: (i Eq. 2 for mass transfer of H 2O to the particle surface; (ii Eq. 13 for mass transfer of H 2O through the product layer; (iii Eq. 21 for mass transfer of H 2O to

form Ca(OH 2 by reaction; and (iv Eq. 18 relating mass transfer of H 2O at the interface of the core and the product layer to experimental data. 6.2 Analysis of Molar Flux through the Product Layer at its Boundaries Considering the molar flux of H 2O at the particle surface, where r r S, Eq. 22 can be derived. Note that r C /r S is related to the conversion to Ca(OH 2 according to Eq. 17. CD e ln [ (1 X S (1 X W H2 O,r s C ] r 2 ( 1 r 1 S r C rr s CD e ln [ (1 X S (1 X C ] r S (1 1 r C /r S Eq. 22 Considering the molar flux of H 2O at the interface of the CaO core and Ca(OH 2 product layer, where r r c, Eq. 23 can be derived. CD e ln [ (1 X S (1 X W H2 O,r c C ] r 2 ( 1 r 1 S r C rr c CD e ln [ (1 X S (1 X C ] r S (( r C rs 2 r C rs Eq. 23 6.3 Finding Solutions for the SCM A solution for k A was found for each set of experimental conditions separately (i.e., for each hydration temperature and number of cycles before hydration. Initially, 41 data points were taken from each experiment at equal conversion increments for conversions to Ca(OH 2 of between 0.05 and either 0.95, if full conversion was achieved, or the final conversion at t 1080 s. Subsequently, X S and X C were calculated using the two equations Eq. 24 and Eq. 25 for all data points, using values of dx Ca(OH2 / and X Ca(OH2 (used to calculate r C /r S with Eq. 17 determined experimentally. X S and X C could not be calculated analytically and, therefore, were established numerically using a

script written in Matlab. Eq. 24 was obtained by substituting Eq. 23 into Eq. 18, while Eq. 25 considers the core of the particles and was derived by equating Eq. 2 and Eq. 22. dx Ca(OH2 k G (X B X S 3(1 X Ca(OH2 2/3 M C D e ln [ (1 X S CaO (1 X C ] ρ CaO (1 ε CaO r 2 S (( r C rs 2 r C rs D e ln [ (1 X S (1 X C ] r S (1 1 (r C /r S Eq. 24 Eq. 25 Then, k A was calculated for each experiment from Eq. 21 by a least squares method across all data points using a script written in Matlab. In this way, a value of k A was established for each set of experimental conditions. The values of k A obtained were then used to obtain projected idealised model results for X S, X C and X Ca(OH2 as a function of time. 7 References 1. Cussler EL, Diffusion: Mass Transfer in Fluid Systems, 3rd Ed. 2009: Cambridge University Press. 2. Perry RH and Green DW, Perry's Chemical Engineers' Handbook, seventh edition. 1997: McGraw-Hill, New York, USA. 3. Lemmon EW, McLinden MO, and Friend DG. Thermophysical Properties of Fluid Systems in NIST Chemistry WebBook, NIST Standard Reference Number 69. Eds. P.J. Linstrom, W.G. Mallard. National Institute of Standards and Technology, Gaithersburg MD, USA. 2009; Available from: http://webbook.nist.gov. 4. Missen RW, Mims CA, and Saville BA, Introduction to Chemical Reaction Engineering and Kinetics. 1999: John Wiley & Sons. 5. Fogler HS, Elements of Chemical Engineering. 2006: Pearson Education International. 6. Bird RB, Stewart WE, and Lightfoot EN, Transport Phenomena, 2nd Ed. 2007: John Wiley & Sons, Inc.